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• agapito
In summary, the author says that Baby Arithmetic does not include the axioms of standard propositional logic, but includes Leibniz's Law and reflexivity of equality.
agapito
I'm going through Peter Smith's book on Godel's Theorems. He mentions a simple formal theory ("Baby Arithmetic") whose logic needs to prove every instance of 'tau = tau'. Does every 'standard deductive apparatus' include the common identity axioms (e.g. 'x = x')?.

The axioms of "Baby Arithmetic" do not include them, so does this mean they are somewhat hidden in the undefined "standard propositional logic" used? He mentions, for instance, that the theory trivially proves '0 = 0'.

The application of Leibniz' Law is explicitly included, however.

Any help greatly appreciated. Agapito

agapito said:
I'm going through Peter Smith's book on Godel's Theorems. He mentions a simple formal theory ("Baby Arithmetic") whose logic needs to prove every instance of 'tau = tau'. Does every 'standard deductive apparatus' include the common identity axioms (e.g. 'x = x')?
One can consider theories with equality and theories without equality. The first ones include the predicate symbol $=$ and axioms saying that equality is a congruence, i.e., an equivalence relation that is respected by all functions and predicates.

agapito said:
The axioms of "Baby Arithmetic" do not include them, so does this mean they are somewhat hidden in the undefined "standard propositional logic" used?
First, in dealing with arithmetic we are working with predicate, or first-order, logic and not propositional logic. Second, the author does include reflexivity of equality. He says, "our logic needs to prove every instance of $\tau = \tau$ , where $\tau$ is a term of our language $\mathcal{L}_B$. And we need a version of Leibniz’s Law". Reflexivity and Leibniz’s Law are treated similarly. These two axioms (or rules of inference; the author leaves the details unspecified because they are not important) are sufficient for proving that equality is a congruence.

Thanks for your thoughtful reply. As I read the text, our logic must prove 'τ=τ', that is we're not using reflexivity as an axiom. We are, however, using Leibniz' Law.

Is it possible therefore to prove '0=0' by only using LL and the 6 axiom schemata of the theory?

Thanks again for helping me out with this.

agapito said:
As I read the text, our logic must prove 'τ=τ', that is we're not using reflexivity as an axiom.
This conclusion is not warranted. One of possible definitions of a derivation (in Hilbert-style axiomatic systems) says: a derivation is a sequence of formulas where each formula is either an axiom or is obtained from two previous formulas by Modus Ponens. That is, axioms are allowed in derivations (of course), and there is no restriction that a derivation cannot end with an axiom. In other words, the fact that $x=x$ is proved does not mean that it is not proved directly by an axiom. In fact, $x=x$ is probably derived from the axiom $\forall x.\,x=x$ using a rule like quantifier elimination or some axiom serving a similar purpose.

agapito said:
Is it possible therefore to prove '0=0' by only using LL and the 6 axiom schemata of the theory?
I don't think so. Reflexivity $\forall x.\,x=x$ and Leibniz's law $\forall x.\,(P(x)\to\forall y\,(x=y\to P(y)))$ are in some sense the opposites of each other: reflexivity says how to prove equality, and LL says how to use it to prove other facts. In the same sense they are similar to pairs of axioms like
\begin{align}
&A\to B\to A\land B\\
&A\land B\to A
\end{align}
and
\begin{align}
&A\to A\lor B\\
&(A\to C)\to(B\to C)\to(A\lor B\to C)
\end{align}
that say how to prove and use conjunction and disjunction, respectively. So I believe both are necessary. But deriving symmetry and transitivity from reflexivity and LL is a nice exercise.

Thanks again for your help, it's very useful. I will attempt the exercise you mention. Agapito

## 1. What are identity axioms in mathematics?

Identity axioms are fundamental principles that define the concept of identity in mathematics. They are a set of assumptions or rules that must hold true for all objects and operations within a mathematical system.

## 2. What is the purpose of identity axioms?

The purpose of identity axioms is to provide a framework for reasoning and problem solving in mathematics. They ensure that mathematical operations are consistent and that the properties of objects remain the same regardless of how they are manipulated.

## 3. What are the basic identity axioms?

The basic identity axioms include the reflexive property, which states that any object is equal to itself; the symmetric property, which states that if two objects are equal, then they can be reversed; and the transitive property, which states that if two objects are equal to a third object, then they are also equal to each other.

## 4. How do identity axioms differ from other axioms?

Identity axioms specifically deal with the concept of identity, while other axioms may address different mathematical concepts such as addition or multiplication. Identity axioms are also considered to be self-evident truths, while other axioms may need to be proven.

## 5. Can identity axioms be violated?

No, identity axioms cannot be violated within a mathematical system. If an operation or object does not adhere to the identity axioms, then it is not considered to be a part of that system. Any mathematical system must adhere to the identity axioms in order to be considered valid.

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